Orbital angular momentum (OAM) is a recently discovered property of free electrons, also called electron vortices.
Recent work has focused both on fundamental questions regarding electron vortex production, dynamics and properties. Work has also been done on techniques for preparing electron probe beams with angular momentum.
A scattered free electron beam from a target results in a distribution of orbital angular momentum states that can be thought of as a new kind of spectrum. The OAM distribution can provide new kinds of information about the structural chirality and out-of-plane magnetization of the target.
Such information would have applications in electron microscopy, spectroscopy, laboratory systems, and at synchrotrons.
In these applications, the electrons can scatter to many different final orbital angular momentum states, and measurement of the final orbital angular momentum distribution could provide new information about the scattering targets.
However, there are currently no measurement techniques that can efficiently and quantitatively measure the orbital angular momentum distribution of free electrons.
Only a few publications to date have discussed measurement of electron orbital angular momentum, and all are based on techniques first developed for photon orbital angular momentum measurement. However, these tecniques will not work for scattered electrons, which are in incoherent mixtures of energy and orbital angular momentum states.
In 2010, Berkhout et al. demonstrated a new method to efficiently sort OAM states of light using four refractive optical elements. The apparatus transforms an azimuthal phase at the input into a linear phase at the output, such that OAM components at the input are mapped into separate linear momentum states at the output. This ability to measure superpositions and mixed states of optical OAM states of light enables parallel orbital angular momentum measurement.
The apparatus has been rapidly employed for a range of optical applications in both fundamental research, quantum information, and communications. As shown in FIG. 1A, the apparatus is based on two custom-made non-spherical refractive optical components, the phase unwrapper 102 (U) and the phase corrector 106 (C), together with two lens systems 104 (L1) and 108 (L2) used to Fourier transform the output of U and C, respectively. Phase unwrapper element U is positioned in the front focal plane of a lens L1. Phase corrector element C is positioned in the back focal plane of L1.
The phase unwrapper 102 (U) and lens system 104 (L1) form a log-polar transformer that transforms a set of concentric rings at the input plane into a set of parallel lines at the back focal plane of the lens—or, equivalently, orbital angular momentum states into planar waves. The corresponding unwrapping phase profile is described by:
                                                        φ              U                        ⁡                          (                              x                ,                y                            )                                =                                    1                              Δ                ⁢                                                                  ⁢                t                                      [                                          y                ⁢                                                                  ⁢                                  arctan                  ⁡                                      (                                          y                      x                                        )                                                              -                              x                ⁢                                                                  ⁢                                  ln                  (                                                                                                              x                          2                                                +                                                  y                          2                                                                                      b                                    )                                            +              x                        ]                          ,                            (        1        )            where Δt is a length scale that sets the separation distance between orbital angular momentum states in the output plane, and b is a length scale that determines the position of the unwrapped light beam in the corrector plane.
A light beam 100 with mixed OAM states is sorted by the device to spatially separate beams 110, 112, 114 having distinct OAM states. In FIG. 1B, light beams 110, 112, 114 with different OAM states are shown in different shades. Immediately after the corrector element 106 (C), different OAM components are separated in momentum space. The Fourier-transforming lens 108 (L2) then separates the OAM components into different beams in position space.
This device for sorting OAM states of light, however, has no straightforward implementation for electrons. In light optics, there are established methods for fabricating custom phase plates out of transparent material such as glass. Although thin film phase plates for electrons are possible, they contaminate easily and are difficult to fabricate. In addition, no material is sufficiently electron-transparent to imprint the large phases required for sorting OAM.
Arbitrary electron phase profiles can be imprinted holographically using nanofabricated diffractive optics. However, the smaller but still significant inelastic scattering in the material, the small diffraction angles, low diffraction efficiency, and finite size of the diffractive structures make the use of such holograms for an OAM mode sorter impractical.
Thus, it remains an unsolved problem to provide a device for measuring and/or sorting OAM states of free electrons.